Comparison.  This result is related to [1]} that studied the phenomenon of benign overfitting in high-dimensional \(\ell _2\)  linear regression. Our result coincides with theirs as both results uncover some scenarios for benign overfitting in linear regression. In particular, [1]} focused on the minimum \(\ell _2\)  norm estimator. They showed that, if the eigenvalue sequence of the covariance operator \(\Sigma := \mathbb {E} [x x^\top ]\)  have suitable decay rates, then the generalization error will decrease to zero as \(n\)  increases. In contrast, our result differs from theirs in the problem settings. Specifically, we do not assume the minimum norm property and consider the optimization process that was omitted in [1]}. Also, we evaluate the generalization error of the convergence point on a globally Lipschitz loss function. In our setting, entries of each input \(x\)  are i.i.d. random variables, meaning that \(\Sigma = \mathbb {I}_{d \times d}\)  without decaying eigenvalues. The requirement of i.i.d. entries for each input is a limitation due to the lack of non-asymptotic results of the spectral properties for random matrices with non-i.i.d. entries. However, this limitation can be overcome if one considers the asymptotic results ([4]}, [5]}). Moreover, our result works for \(\ell _p\)  loss functions with any even positive integer \(p\) . This expands the understanding of benign overfitting in the high-dimensional setting.
